CYLINDER (Gr. κύλινδρος, from κυλίνδειν, to roll). A cylindrical surface, or briefly a cylinder, is the surface traced out by a line, named the generatrix, which moves parallel to itself and always passes through the circumference of a curve, named the directrix; the name cylinder is also given to the solid contained between such a surface and two parallel planes which intersect a generatrix. A “right cylinder” is the solid traced out by a rectangle which revolves about one of its sides, or the curved surface of this solid; the surface may also be defined as the locus of a line which passes through the circumference of a circle, and is always perpendicular to the plane of the circle. If the moving line be not perpendicular to the plane of the circle, but moves parallel to itself, and always passes through the circumference, it traces an “oblique cylinder.” The “axis” of a circular cylinder is the line joining the centres of two circular sections; it is the line through the centre of the directrix parallel to the generators. The characteristic property of all cylindrical surfaces is that the tangent planes are parallel to the axis. They are “developable” surfaces, i.e. they can be applied to a plane surface without crinkling or tearing (see Surface).

Any section of a cylinder which contains the axis is termed a “principal section”; in the case of the solids this section is a rectangle; in the case of the surfaces, two parallel straight lines. A section of the right cylinder parallel to the base is obviously a circle; any other section, excepting those limited by two generators, is an ellipse. This last proposition may be stated in the form: — “The orthogonal projection of a circle is an ellipse”; and it permits the ready deduction of many properties of the ellipse from the circle. The section of an oblique cylinder by a plane perpendicular to the principal section, and inclined to the axis at the same angle as the base, is named the “subcontrary section,” and is always a circle; any other section is an ellipse.

The mensuration of the cylinder was worked out by Archimedes, who showed that the volume of any cylinder was equal to the product of the area of the base into the height of the solid, and that the area of the curved surface was equal to that of a rectangle having its sides equal to the circumference of the base, and to the height of the solid. If the base be a circle of radius r, and the height h, the volume is πr2h and the area of the curved surface 2πrh. Archimedes also deduced relations between the sphere (q.v.) and cone (q.v.) and the circumscribing cylinder.

The name “cylindroid” has been given to two different surfaces. Thus it is a cylinder having equal and parallel elliptical bases; i.e. the surface traced out by an ellipse moving parallel to itself so that every point passes along a straight line, or by a line moving parallel to itself and always passing through the circumference of a fixed ellipse. The name was also given by Arthur Cayley to the conoidal cubic surface which has for its equation z(x2 + y2) = 2mxy; every point on this surface lies on the line given by the intersection of the planes y = x tan θ, z = m sin 2θ, for by eliminating θ we obtain the equation to the surface.